The dispersion curve of lattice have two branches the acoustical branch and the optical branch. The acoustical branch is so called because the dispersion curve is like that of sound wave (frequency increases with wave number ) while optical branch,s dispersion curve is little depending on the wave number .

Einstein model is adequate for interpreting the optical branch while Debey model success in explaining the acoustical branch. principle of solid state physics by Levey is an excellent text for explaining this topic.

regards

Einstein model neglects the dispersion of phonon frequencies , i.e., the phonons are represented as a system of independent quantum oscillators of equal frequency. The theory is satisfactory for high temperatures, but for low temperatures it fails because it implies the Dulong-Petit law independent of the temperature;That is not the case of low temperatures...

The Debye theory respects the dependence of the phonon frequencies on the wave vector: it respects the discreteness of the lattice. As a result, it yields correct dependence of the heat capacity for low temperatures ~T^3. However, the theory is constructed for phonons as a system of linear oscillators, so that it does not include the lattice anharmonicity.

The dispersion curve of lattice have two branches the acoustical branch and the optical branch. The acoustical branch is so called because the dispersion curve is like that of sound wave (frequency increases with wave number ) while optical branch,s dispersion curve is little depending on the wave number .

Einstein model is adequate for interpreting the optical branch while Debey model success in explaining the acoustical branch. principle of solid state physics by Levey is an excellent text for explaining this topic.

regards

Yes, You are right, but of course, the optical branches obey their dispersion law as well.

Actually, the phonon is a kind of elemental excitation (or quasi particle). While the Debye or Einstein model are designed to treat the thermodynamic properties of periodic crystal. Maybe you want to solve the thermodynamic problems. I encourage you to read the relative papers of SCAILD, which can include the anharmonic effects at high temperature.

Neither Einstein or Debye model is correct but these are useful to derive approximate analytical expressions for specific heat and other thermodynamic quantities. To do better one has to calculate the phonon density of states. The complete information about phonons in a crystal consists of the dispersions of all acoustic and optic modes and their structure factors and the phonon density of states. There exist several programs to do ab-initio calculations of all these quantities.

The Debye theory introduced the density of state to derive expression of specific heat and that is a good agreement with the experimental.

The debye approximation for phonon the density of states is crude but the thermodynamic functions are calculated from its moments and is not very sensitive to the details of it.

For ab-initio calculations of phonon density of states in LaMnO3 and ReO3 see our papers in Research Gate.

Ab-initio phonon calculations have become very important in condensed matter physics and materials science. However one must first calculate electron structure (DFT method) and optimize the relevant potentials. You can then proceed to calculate phonon by some available codes. Parlinski in Krakaw has a very good phonon code but one has to pay for it. But I guess there are free soft wares as well. The ab-initio phonon dispersion calculations should be checked with Raman and infrared measurements for the zone center frequencies and then the calculated phonon dispersions should be checked with the inelastic neutron scattering results if available. One should also calculate the phonon density of states and this should also be compared with inelastic neutron scattering measurements. If the agreement is reasonable then you know that your calculations are right and successful. You can of course then calculate all thermodynamic quantities from the phonon density of states. But as I mentioned before these thermodynamic quantities like specific heat etc. is not very sensitive to the details of the phonon density of states. That is why Einstein (only limited success) and especially the Debye theory the phonon contribution to the specific heat is more or less successful for simple cases.

Dear Chatterji and others,

When you mention the Parlinski code I suppose that you only have DFT codes that only calculate force constants, like vasp or wien2k, by the force constant method. I have my suspictions about the Parlinski code once it does not take into account the vibration of effective charges that are responsible for the LO-TO splitting. Instead, I suggested to use modern codes to calculate directly the correct phonon bands and density of states, like abinit or espresso, without the help of the Parlinski code. By the way, both Einstein and Debye's model are crude ones to describe the phonons. However, the Debye one is mostly used to describe thermodynamic quantities with relatively success. But, if you can obtain the correct density of phonon states, you can calculate these thermodynamic functions simply by integrating the partition function and I have done them for the III-Nitrides as well as for polymers: look for L. S. Pereira et al, http://dx.doi.org/10.1016/S0026-2692(03)00088-0 (III-Nitrides) and; R. L. Sousa and H. W. Leite Alves, http://dx.doi.org/10.1590/S0103-97332006000300072 (PPP and PPV polymers).

The best method for calculating heat capacity is to calculate it from the experiment or calculated (ab-initio) phonon density of states. If the sample is magnetic then there will be magnetic contribution as well. I have outlined the method before.

Think also on the lattice temperature you're considering and it's value relative to the Debye temperature. At higher temperatures the Debye approach collapses into Einstein's.

Neither is reliable. Physically, the two models describe different excitations of the lattice: the Debye model describes the acoustic phonons, which are gapless excitations (they are thus very relevant to the thermodynamic properties at very low-temperatures), and the Einstein model the optical phonons, which are gapped excitations. These models can best be understood in the framework of the so-called 'problem of moments': given a distribution function (here the density of the phonon states -- technically, any function that is either positive, or negative, and is integrable over its entire support), the problem is whether it can be described in terms of its moments. One then enters into the realm of the continued-fraction expansion of distribution functions and the considerations regarding truncation of the continued-fraction expansions of these functions. For this subject matter, the usual starting point is the very admirable monograph 'The Problem of Moments', by Shohat and Tamarkin (American Mathematical Society, 4th printing of the revised edition, 1970).

Neither is reliable seperately. In my opinion both- the combined model so called, 'Debye-Einstein model' is the most reliable model for Phonons. This model accounts acoustic phonons at low-temperatures relating to the  Debye temperature and the optical phonons relating to the Debye analogy of Einstein temperature. Such concept has been reliably adopted in specific heat, Pyroelectric effect and may be in thermal conductivity as now it is realized that optical phonons in thin film of nano size contribute to the thermal conductivity at low temperatures. Although the well accepted Callaway theory neglects the contribution of optical phonons, still its open to include it. 

Einstein’s theory of heat capacities: Einstein treated the atoms in a crystal as N simple harmonic oscillators, all having the same frequency νE. The frequency νE depends on the strength of the restoring force acting on the atom, i.e. the strength of the chemical bonds within the solid.

θE is the ‘Einstein temperature’, which is different for each solid, and reflects the rigidity of the

lattice. At the high temperature limit, when T >> θE (and x 1), Cv = 0 as T = 0, as required by the third law of

thermodynamics. [Prove by setting ex-1 ~ ex in the denominator for large x].

Debye’s theory of heat capacities

Debye improved on Einstein’s theory by treating the coupled vibrations of the solid in terms of 3N

normal modes of vibration of the whole system, each with its own frequency. The lattice vibrations

are therefore equivalent to 3N independent harmonic oscillators with these normal mode frequencies.

For low frequency vibrations, defined as those for which the wavelength is much greater than the

atomic spacing, λ >> a, the crystal may be treated as a homogeneous elastic medium. The normal modes are the frequencies of the standing waves that are possible in the medium?

The Debye heat capacity depends only on the Debye temperature θD.

The integral cannot be evaluate At high temperatures (T >> θD, xD 1), we note that the integrand tends towards

zero rapidly for large x. This allows us to replace the upper limit by ∞ and turn the integral into a

standard integral, to give Cv = 3Nk

We see that the Debye heat capacity decreases as T3 at low temperatures, in agreement with

experimental observation. This is a marked improvement on Einstein’s theory.ted analytically, but the bracketed function is tabulated

to find that at low temperatures TD/T → ∞, so the behavior is like that of the photon gas in d dimensions, with c → vs, and multiplied by d/2 due to the difference in degeneracy (2 for photons, d for phonons)

At high temperatures, the behavior is like that of a classical gas in a harmonic potential so that U = dN kBT, CV = N dkB, and P V = 2U/d

Debye model for phonons and Einstein model for optical modes

Debye model is more reliable than Einstein model.